Theorem filuni 22197

Description: The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
filuni	⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → ∪ 𝐹 ∈ (Fil‘𝑋))

Distinct variable groups:   𝑓,𝑔,𝐹   𝑓,𝑋,𝑔

Proof of Theorem filuni

Dummy variables ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4716	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐹 ↔ ∃𝑓 ∈ 𝐹 𝑥 ∈ 𝑓)
2		ssel2 3853	. . . . . . 7 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑓 ∈ 𝐹) → 𝑓 ∈ (Fil‘𝑋))
3		filelss 22164	. . . . . . . 8 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝑓) → 𝑥 ⊆ 𝑋)
4	3	ex 405	. . . . . . 7 ⊢ (𝑓 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝑓 → 𝑥 ⊆ 𝑋))
5	2, 4	syl 17	. . . . . 6 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑓 ∈ 𝐹) → (𝑥 ∈ 𝑓 → 𝑥 ⊆ 𝑋))
6	5	rexlimdva 3229	. . . . 5 ⊢ (𝐹 ⊆ (Fil‘𝑋) → (∃𝑓 ∈ 𝐹 𝑥 ∈ 𝑓 → 𝑥 ⊆ 𝑋))
7	6	3ad2ant1 1113	. . . 4 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → (∃𝑓 ∈ 𝐹 𝑥 ∈ 𝑓 → 𝑥 ⊆ 𝑋))
8	1, 7	syl5bi 234	. . 3 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → (𝑥 ∈ ∪ 𝐹 → 𝑥 ⊆ 𝑋))
9	8	pm4.71rd 555	. 2 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → (𝑥 ∈ ∪ 𝐹 ↔ (𝑥 ⊆ 𝑋 ∧ 𝑥 ∈ ∪ 𝐹)))
10		ssn0 4240	. . . 4 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅) → (Fil‘𝑋) ≠ ∅)
11		fvprc 6492	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (Fil‘𝑋) = ∅)
12	11	necon1ai 2994	. . . 4 ⊢ ((Fil‘𝑋) ≠ ∅ → 𝑋 ∈ V)
13	10, 12	syl 17	. . 3 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝑋 ∈ V)
14	13	3adant3 1112	. 2 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → 𝑋 ∈ V)
15		filtop 22167	. . . . . . . . 9 ⊢ (𝑓 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝑓)
16	2, 15	syl 17	. . . . . . . 8 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑓 ∈ 𝐹) → 𝑋 ∈ 𝑓)
17	16	a1d 25	. . . . . . 7 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑓 ∈ 𝐹) → (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 → 𝑋 ∈ 𝑓))
18	17	ralimdva 3127	. . . . . 6 ⊢ (𝐹 ⊆ (Fil‘𝑋) → (∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 → ∀𝑓 ∈ 𝐹 𝑋 ∈ 𝑓))
19		r19.2z 4323	. . . . . . 7 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 𝑋 ∈ 𝑓) → ∃𝑓 ∈ 𝐹 𝑋 ∈ 𝑓)
20	19	ex 405	. . . . . 6 ⊢ (𝐹 ≠ ∅ → (∀𝑓 ∈ 𝐹 𝑋 ∈ 𝑓 → ∃𝑓 ∈ 𝐹 𝑋 ∈ 𝑓))
21	18, 20	sylan9 500	. . . . 5 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅) → (∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 → ∃𝑓 ∈ 𝐹 𝑋 ∈ 𝑓))
22	21	3impia 1097	. . . 4 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → ∃𝑓 ∈ 𝐹 𝑋 ∈ 𝑓)
23		eluni2 4716	. . . 4 ⊢ (𝑋 ∈ ∪ 𝐹 ↔ ∃𝑓 ∈ 𝐹 𝑋 ∈ 𝑓)
24	22, 23	sylibr 226	. . 3 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → 𝑋 ∈ ∪ 𝐹)
25		sbcel1v 3742	. . 3 ⊢ ([𝑋 / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ 𝑋 ∈ ∪ 𝐹)
26	24, 25	sylibr 226	. 2 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → [𝑋 / 𝑥]𝑥 ∈ ∪ 𝐹)
27		0nelfil 22161	. . . . . 6 ⊢ (𝑓 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝑓)
28	2, 27	syl 17	. . . . 5 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑓 ∈ 𝐹) → ¬ ∅ ∈ 𝑓)
29	28	ralrimiva 3132	. . . 4 ⊢ (𝐹 ⊆ (Fil‘𝑋) → ∀𝑓 ∈ 𝐹 ¬ ∅ ∈ 𝑓)
30	29	3ad2ant1 1113	. . 3 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → ∀𝑓 ∈ 𝐹 ¬ ∅ ∈ 𝑓)
31		sbcel1v 3742	. . . . . 6 ⊢ ([∅ / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ ∅ ∈ ∪ 𝐹)
32		eluni2 4716	. . . . . 6 ⊢ (∅ ∈ ∪ 𝐹 ↔ ∃𝑓 ∈ 𝐹 ∅ ∈ 𝑓)
33	31, 32	bitri 267	. . . . 5 ⊢ ([∅ / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ ∃𝑓 ∈ 𝐹 ∅ ∈ 𝑓)
34	33	notbii 312	. . . 4 ⊢ (¬ [∅ / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ ¬ ∃𝑓 ∈ 𝐹 ∅ ∈ 𝑓)
35		ralnex 3183	. . . 4 ⊢ (∀𝑓 ∈ 𝐹 ¬ ∅ ∈ 𝑓 ↔ ¬ ∃𝑓 ∈ 𝐹 ∅ ∈ 𝑓)
36	34, 35	bitr4i 270	. . 3 ⊢ (¬ [∅ / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ ∀𝑓 ∈ 𝐹 ¬ ∅ ∈ 𝑓)
37	30, 36	sylibr 226	. 2 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → ¬ [∅ / 𝑥]𝑥 ∈ ∪ 𝐹)
38		simp13 1185	. . . . 5 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) → ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹)
39		r19.29 3200	. . . . . 6 ⊢ ((∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ ∃𝑓 ∈ 𝐹 𝑥 ∈ 𝑓) → ∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓))
40	39	ex 405	. . . . 5 ⊢ (∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 → (∃𝑓 ∈ 𝐹 𝑥 ∈ 𝑓 → ∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓)))
41	38, 40	syl 17	. . . 4 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) → (∃𝑓 ∈ 𝐹 𝑥 ∈ 𝑓 → ∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓)))
42		simp1 1116	. . . . 5 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → 𝐹 ⊆ (Fil‘𝑋))
43		simp1 1116	. . . . . . . . 9 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) → 𝐹 ⊆ (Fil‘𝑋))
44		simpl 475	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝐹 ∧ (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓)) → 𝑓 ∈ 𝐹)
45	43, 44, 2	syl2an 586	. . . . . . . 8 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) ∧ (𝑓 ∈ 𝐹 ∧ (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓))) → 𝑓 ∈ (Fil‘𝑋))
46		simprrr 769	. . . . . . . 8 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) ∧ (𝑓 ∈ 𝐹 ∧ (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓))) → 𝑥 ∈ 𝑓)
47		simpl2 1172	. . . . . . . 8 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) ∧ (𝑓 ∈ 𝐹 ∧ (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓))) → 𝑦 ⊆ 𝑋)
48		simpl3 1173	. . . . . . . 8 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) ∧ (𝑓 ∈ 𝐹 ∧ (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓))) → 𝑥 ⊆ 𝑦)
49		filss 22165	. . . . . . . 8 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝑓 ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦)) → 𝑦 ∈ 𝑓)
50	45, 46, 47, 48, 49	syl13anc 1352	. . . . . . 7 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) ∧ (𝑓 ∈ 𝐹 ∧ (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓))) → 𝑦 ∈ 𝑓)
51	50	expr 449	. . . . . 6 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) ∧ 𝑓 ∈ 𝐹) → ((∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓) → 𝑦 ∈ 𝑓))
52	51	reximdva 3219	. . . . 5 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) → (∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓) → ∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓))
53	42, 52	syl3an1 1143	. . . 4 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) → (∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑓) → ∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓))
54	41, 53	syld 47	. . 3 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) → (∃𝑓 ∈ 𝐹 𝑥 ∈ 𝑓 → ∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓))
55		sbcel1v 3742	. . . 4 ⊢ ([𝑥 / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ 𝑥 ∈ ∪ 𝐹)
56	55, 1	bitri 267	. . 3 ⊢ ([𝑥 / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ ∃𝑓 ∈ 𝐹 𝑥 ∈ 𝑓)
57		sbcel1v 3742	. . . 4 ⊢ ([𝑦 / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ 𝑦 ∈ ∪ 𝐹)
58		eluni2 4716	. . . 4 ⊢ (𝑦 ∈ ∪ 𝐹 ↔ ∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓)
59	57, 58	bitri 267	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ ∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓)
60	54, 56, 59	3imtr4g 288	. 2 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑦) → ([𝑥 / 𝑥]𝑥 ∈ ∪ 𝐹 → [𝑦 / 𝑥]𝑥 ∈ ∪ 𝐹))
61		simp13 1185	. . . . 5 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹)
62		r19.29 3200	. . . . . 6 ⊢ ((∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ ∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓) → ∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑦 ∈ 𝑓))
63	62	ex 405	. . . . 5 ⊢ (∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 → (∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓 → ∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑦 ∈ 𝑓)))
64	61, 63	syl 17	. . . 4 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → (∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓 → ∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑦 ∈ 𝑓)))
65		simp11 1183	. . . . 5 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ (Fil‘𝑋))
66		r19.29 3200	. . . . . . . . . . 11 ⊢ ((∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ ∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔) → ∃𝑔 ∈ 𝐹 ((𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑔))
67	66	ex 405	. . . . . . . . . 10 ⊢ (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 → (∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔 → ∃𝑔 ∈ 𝐹 ((𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑔)))
68		elun1 4041	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑓 → 𝑦 ∈ (𝑓 ∪ 𝑔))
69		elun2 4042	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑔 → 𝑥 ∈ (𝑓 ∪ 𝑔))
70	68, 69	anim12i 603	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑓 ∧ 𝑥 ∈ 𝑔) → (𝑦 ∈ (𝑓 ∪ 𝑔) ∧ 𝑥 ∈ (𝑓 ∪ 𝑔)))
71		eleq2 2854	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑓 ∪ 𝑔) → (𝑦 ∈ ℎ ↔ 𝑦 ∈ (𝑓 ∪ 𝑔)))
72		eleq2 2854	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑓 ∪ 𝑔) → (𝑥 ∈ ℎ ↔ 𝑥 ∈ (𝑓 ∪ 𝑔)))
73	71, 72	anbi12d 621	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓 ∪ 𝑔) → ((𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ) ↔ (𝑦 ∈ (𝑓 ∪ 𝑔) ∧ 𝑥 ∈ (𝑓 ∪ 𝑔))))
74	73	rspcev 3535	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∪ 𝑔) ∈ 𝐹 ∧ (𝑦 ∈ (𝑓 ∪ 𝑔) ∧ 𝑥 ∈ (𝑓 ∪ 𝑔))) → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ))
75	70, 74	sylan2 583	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∪ 𝑔) ∈ 𝐹 ∧ (𝑦 ∈ 𝑓 ∧ 𝑥 ∈ 𝑔)) → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ))
76	75	an12s 636	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑓 ∧ ((𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑔)) → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ))
77	76	ex 405	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑓 → (((𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑔) → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ)))
78	77	ad2antlr 714	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ 𝐹 ∧ 𝑦 ∈ 𝑓) ∧ 𝑔 ∈ 𝐹) → (((𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑔) → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ)))
79	78	rexlimdva 3229	. . . . . . . . . 10 ⊢ ((𝑓 ∈ 𝐹 ∧ 𝑦 ∈ 𝑓) → (∃𝑔 ∈ 𝐹 ((𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑥 ∈ 𝑔) → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ)))
80	67, 79	syl9r 78	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝐹 ∧ 𝑦 ∈ 𝑓) → (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 → (∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔 → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ))))
81	80	impr 447	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝐹 ∧ (𝑦 ∈ 𝑓 ∧ ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹)) → (∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔 → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ)))
82	81	ancom2s 637	. . . . . . 7 ⊢ ((𝑓 ∈ 𝐹 ∧ (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑦 ∈ 𝑓)) → (∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔 → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ)))
83	82	rexlimiva 3226	. . . . . 6 ⊢ (∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑦 ∈ 𝑓) → (∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔 → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ)))
84	83	imp 398	. . . . 5 ⊢ ((∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑦 ∈ 𝑓) ∧ ∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔) → ∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ))
85		ssel2 3853	. . . . . . 7 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ ℎ ∈ 𝐹) → ℎ ∈ (Fil‘𝑋))
86		filin 22166	. . . . . . . 8 ⊢ ((ℎ ∈ (Fil‘𝑋) ∧ 𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ) → (𝑦 ∩ 𝑥) ∈ ℎ)
87	86	3expib 1102	. . . . . . 7 ⊢ (ℎ ∈ (Fil‘𝑋) → ((𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ) → (𝑦 ∩ 𝑥) ∈ ℎ))
88	85, 87	syl 17	. . . . . 6 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ ℎ ∈ 𝐹) → ((𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ) → (𝑦 ∩ 𝑥) ∈ ℎ))
89	88	reximdva 3219	. . . . 5 ⊢ (𝐹 ⊆ (Fil‘𝑋) → (∃ℎ ∈ 𝐹 (𝑦 ∈ ℎ ∧ 𝑥 ∈ ℎ) → ∃ℎ ∈ 𝐹 (𝑦 ∩ 𝑥) ∈ ℎ))
90	65, 84, 89	syl2im 40	. . . 4 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → ((∃𝑓 ∈ 𝐹 (∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹 ∧ 𝑦 ∈ 𝑓) ∧ ∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔) → ∃ℎ ∈ 𝐹 (𝑦 ∩ 𝑥) ∈ ℎ))
91	64, 90	syland 593	. . 3 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → ((∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓 ∧ ∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔) → ∃ℎ ∈ 𝐹 (𝑦 ∩ 𝑥) ∈ ℎ))
92		eluni2 4716	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐹 ↔ ∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔)
93	55, 92	bitri 267	. . . 4 ⊢ ([𝑥 / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ ∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔)
94	59, 93	anbi12i 617	. . 3 ⊢ (([𝑦 / 𝑥]𝑥 ∈ ∪ 𝐹 ∧ [𝑥 / 𝑥]𝑥 ∈ ∪ 𝐹) ↔ (∃𝑓 ∈ 𝐹 𝑦 ∈ 𝑓 ∧ ∃𝑔 ∈ 𝐹 𝑥 ∈ 𝑔))
95		sbcel1v 3742	. . . 4 ⊢ ([(𝑦 ∩ 𝑥) / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ (𝑦 ∩ 𝑥) ∈ ∪ 𝐹)
96		eluni2 4716	. . . 4 ⊢ ((𝑦 ∩ 𝑥) ∈ ∪ 𝐹 ↔ ∃ℎ ∈ 𝐹 (𝑦 ∩ 𝑥) ∈ ℎ)
97	95, 96	bitri 267	. . 3 ⊢ ([(𝑦 ∩ 𝑥) / 𝑥]𝑥 ∈ ∪ 𝐹 ↔ ∃ℎ ∈ 𝐹 (𝑦 ∩ 𝑥) ∈ ℎ)
98	91, 94, 97	3imtr4g 288	. 2 ⊢ (((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) ∧ 𝑦 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → (([𝑦 / 𝑥]𝑥 ∈ ∪ 𝐹 ∧ [𝑥 / 𝑥]𝑥 ∈ ∪ 𝐹) → [(𝑦 ∩ 𝑥) / 𝑥]𝑥 ∈ ∪ 𝐹))
99	9, 14, 26, 37, 60, 98	isfild 22170	1 ⊢ ((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → ∪ 𝐹 ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2967  ∀wral 3088  ∃wrex 3089  Vcvv 3415  [wsbc 3681   ∪ cun 3827   ∩ cin 3828   ⊆ wss 3829  ∅c0 4178  ∪ cuni 4712  ‘cfv 6188  Filcfil 22157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fv 6196  df-fbas 20244  df-fil 22158

This theorem is referenced by:  filssufilg  22223